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We propose a refinement in the interpolative approach in fixed-point theory. In particular, using this method, we prove the existence of fixed points and common fixed points for Kannan-type contractions and provide examples to support our results.
Kannan fixed-point theorem is the first significant variant of the outstanding result of Banach on the metric fixed-point theory [1,2]. Kannan’s theorem has been generalized in different ways. In the present note, we zoom in on one of the recent generalizations that was proposed by Karapınar  as interpolative Kannan-type contraction. It was indicated in  that each interpolative Kannan-type contraction in a complete metric space admits a fixed point (see also e.g., [4,5,6,7]). More precisely, we have:
Let be a complete metric space and an interpolative Kannan-type contraction, i.e., T is a self-map such that there exist with
for all , where .
Then T has a fixed point in X.
Our contribution in the present manuscript aims at sharpening the inequality (1) by increasing the degree of freedom of the powers appearing in the right-hand side in the framework of standard metric spaces. We also indicate the novelty of our results by expressing some examples.
2. Main Results
We start with the following definition.
Let a metric space and a self-map. We shall call T a -interpolative Kannan contraction, if there exist with such that
for all with
We are now ready to state the main result of this paper.
Let a complete metric space and be a -interpolative Kannan contraction with so that . Then T has a fixed point in X.
Following the steps of the proof of (, Theorem 2.2), we construct the sequence by iterating where is an arbitrary starting point. Then, we observe that
As already elaborated in the proof of (, Theorem 2.2), the classical procedure leads to the existence of a unique fixed point □
We conclude this section by presenting an example explaining why our approach is more general.
(Compare , Example 2.3)).Take and endow it with the following metric:
We also define the self-map T on X as
We observe that the inequality:
is satisfied for:
In all these cases, i.e., and the map obviously has a unique fixed point.
In other words, the inequality
could just be replaced by the existence of two reals such that
Inspired by the above question, we introduce the idea of “optimal interpolative triplet ” for a -interpolative Kannan contraction.
Let be a metric space and be a self-map. We shall call T a relaxed -interpolative Kannan contraction, if there exist such that
Let be a metric space and be a relaxed -interpolative Kannan contraction. The triplet will be called “optimal interpolative triplet” if for any , the inequality (3) fails for at least one of the triplet
Therefore, we formulate the following conjecture for which we currently do not have any proof.
Let be a complete metric space. Let be a map such that for any , admits an optimal interpolative triplet . If and , then T has a unique fixed point. Moreover, this fixed point can be obtained via the Picard iteration.
Theorem 2 can easily be generalized to the case of two maps. More precisely:
Let be a metric space and be two self-maps. We shall call a -interpolative Kannan contraction pair, if there exist with such that
for all with
Our result then goes as follows:
Let be a complete metric space and be a -interpolative Kannan contraction pair. Then R and T have a common fixed point in X, i.e., there exists such that .
We construct the sequence by iterating
where is an arbitrary starting point.
The proof then follows the same steps as (, Theorem 2.1). As already elaborated in the proof of (, Theorem 2.1), the classical procedure leads to the existence of a unique fixed point □
We use the metric defined in Example 1. We also define on X the self-maps T as
and R as
We observe that the inequality:
is satisfied for:
R and T have two common fixed points x and
The above conjecture (Theorem 3) motives us in the investigation of interpolative Kannan contraction for a family of maps. Indeed Noorwali  used interpolation to obtain a common fixed-point result for a Kannan-type contraction mapping. We aim at generalizing (, Theorem 2.1) and Theorem 4 with the use of a -interpolative Kannan contraction for a family of maps. More precisely:
Let be a complete metric space. Let be a family of self-maps such for any
What are the conditions on for to have a (unique)common fixed point.
Y.U.G. writing–original draft preparation; E.K. writing–review and editing.
This research received no external funding.
The authors thanks anonymous referees for their remarkable comments, suggestion, and ideas that help to improve this paper. Y.U.G. wishes to thank the African Institute for Mathematical Sciences (AIMS), in South Africa, which accepted him as a visitor in May 2019 and provided full funding for his stay.
Conflicts of Interest
The authors declare no conflict of interest.
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